Grinding is doing the same mission over and over again in order to get some reward, item, experience points, or gear. For example, some of the ships in the C-store can be purchased with in-game currency (emblems when you hit max level). It takes 500 emblems to buy them. So you 'grind' by doing the same missions each day to earn more emblems to buy them. But if this is your first time playing, you will not be grinding for a while. There's plenty of missions and stuff to do to get you leveled up to Rear Admiral.

Only if the que pops. There is not much PVP going on at lower ranks. Not much XP if you take into account the amount of time it takes. I leveled a toon up solely through space PVP. It took forever.

I didn't start PvPing with my VA until I hit VA, when I started my sci character I did some lower level PVP and while I didn't rocket to the top I did go a tad faster. At the lower levels any XP is helpful to get you to the top faster.

I have found that the following lineup is one of the best ways to level

After the tutorial:

activate an 8 hour xp boost
Follow the story line and do the siruis and regulus sector patrol missions
Generally by the time you get half way through the regulus missions your XP bonus will run out and you will graduate to Lt. Commander

Activate another 8 hour XP boost
At this point I do the Devidian and Breen Missions followed by the Klingon Front Missions and Pi Canis Patrol Missions;

This generally gets you to Commander at the start of the Romulan Front Missions which gets you to a point
where you can level normally and fairly quickly

Just what I've found...there are many ways to augment this also:
Fleet Actions, Exploration Missions, PVP, War Zones, etc...